Yup, blackbirds/Magpies will randomly just decide to hate certain cats, a couple blackbirds would follow around our ickle cat where ever she went, they would follow my cat down the road fence to fence squawking at her for no reason. More recently we had to magpies decide to do this to all the cats in our road which was ruddy irritating as they are incredibly loud. Seems those two decided to set up a nest on a holly tree in our garden then annoy everything that got close to it.
Most cats don't bother going after big birds, if a daft bird lands on the floor near them they'll have a go but not really seem to care if they fly away, these magpies just harasse the cats. Was stupidly funny, I looked out the window and saw my bigger cat flopped over on the lawn , and a magpie just went and stood about a metre away basically shouting at my cat, who kept rolling around to get slightly further away and rolled over about 10 metre's before getting annoyed and headed back inside our house.
Some birds I'd be quite happy to be shot for the sake of our sanity. We keep our cats in from around 10pm-9am or so, other neighbours don't but due to these magpies had to start catching them and keeping them in or the magpies would start following/making a lot of noise as soon as it got light and it was waking up everyone in 7-8 houses down the road.
I'd put in a call to a bird protection type society though, see what they recommend, waiting a while as they'll likely move on when the birds leave the nest, or look around for a pest control type company that might deal with birds, don't know if any companies do or can do things to birds.